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Unformatted text preview: LECTURE NOTES MEASURE THEORY and PROBABILITY Rodrigo Ba˜nuelos Department of Mathematics Purdue University West Lafayette, IN 47907 June 20, 2003 2 I SIGMA ALGEBRAS AND MEASURES § 1 σ –Algebras: Definitions and Notation. We use Ω to denote an abstract space. That is, a collection of objects called points. These points are denoted by ω . We use the standard notation: For A, B ⊂ Ω, we denote A ∪ B their union, A ∩ B their intersection, A c the complement of A , A \ B = A B = { x ∈ A : x 6∈ B } = A ∩ B c and A Δ B = ( A \ B ) ∪ ( B \ A ). If A 1 ⊂ A 2 , . . . and A = ∪ ∞ n =1 A n , we will write A n ↑ A . If A 1 ⊃ A 2 ⊃ . . . and A = ∩ ∞ n =1 A n , we will write A n ↓ A . Recall that ( ∪ n A n ) c = ∩ n A c n and ( ∩ n A n ) c = ∪ n A c n . With this notation we see that A n ↑ A ⇒ A c n ↓ A c and A n ↓ A ⇒ A c n ↑ A c . If A 1 , . . . , A n ∈ Ω, we can write ∪ n j =1 A j = A 1 ∪ ( A c 1 ∩ A 2 ) ∪ ( A c 1 ∩ A c 2 ∩ A 3 ) ∪ . . . ( A c 1 ∩ . . . ∩ A c n 1 ∩ A n ) , (1.1) which is a disjoint union of sets. In fact, this can be done for infinitely many sets: ∪ ∞ n =1 A n = ∪ ∞ n =1 ( A c 1 ∩ . . . ∩ A c n 1 ∩ A n ) . (1.2) If A n ↑ , then ∪ n j =1 A j = A 1 ∪ ( A 2 \ A 1 ) ∪ ( A 3 \ A 2 ) . . . ∪ ( A n \ A n 1 ) . (1.3) Two sets which play an important role in studying convergence questions are: lim A n ) = lim sup n A n = ∞ n =1 ∞ [ k = n A k (1.4) and lim A n = lim inf n A n = ∞ [ n =1 ∞ k = n A k . (1.5) 3 Notice ( lim A n ) c = ∞ n =1 ∞ [ k = n A n ! c = ∞ [ n =1 ∞ [ k = n A k ! c = ∞ [ n =1 ∞ k = n A c k = lim A c n Also, x ∈ lim A n if and only if x ∈ ∞ S k = n A k for all n . Equivalently, for all n there is at least one k > n such that x ∈ A k . That is, x ∈ A n for infinitely many n . For this reason when x ∈ lim A n we say that x belongs to infinitely many of the A n s and write this as x ∈ A n i.o. If x ∈ lim A n this means that x ∈ ∞ T k = n A k for some n or equivalently, x ∈ A k for all k > n . For this reason when x ∈ lim A n we say that x ∈ A n , eventually . We will see connections to lim x k , lim x k , where { x k } is a sequence of points later. Definition 1.1. Let F be a collection of subsets of Ω. F is called a field (algebra) if Ω ∈ F and F is closed under complementation and finite union. That is, (i) Ω ∈ F (ii) A ∈ F ⇒ A c ∈ F (ii) A 1 , A 2 , . . . A n ∈ F ⇒ n S j =1 A j ∈ F . If in addition, (iii) can be replaced by countable unions, that is if (iv) A 1 , . . . A n , . . . ∈ F ⇒ ∞ S j =1 A j ∈ F , then F is called a σ –algebra or often also a σ –field ....
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This note was uploaded on 01/02/2012 for the course FINANCE 347 taught by Professor Bayou during the Fall '11 term at NYU.
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